derivative of y=ln x/x^2
can someone double check the answer
1-2lnx/x^3
To find the derivative of the function y = ln(x) / x^2, we can use the quotient rule. The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), then its derivative is given by:
f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2
In this case, g(x) is ln(x) and h(x) is x^2. Let's differentiate each term separately:
g'(x) = 1/x (since the derivative of ln(x) is 1/x)
h'(x) = 2x (since the derivative of x^2 is 2x)
Now, we can substitute these values into the quotient rule formula:
f'(x) = (x^2 * 1/x - ln(x) * 2x) / (x^2)^2
= (x - 2xln(x)) / x^4
= x(1 - 2ln(x)) / x^4
= (1 - 2ln(x)) / x^3
So, the derivative of y = ln(x) / x^2 is (1 - 2ln(x)) / x^3.
Double checking our answer, we can use an online derivative calculator or software to find the derivative, and we should obtain the same result.